Total energy of the system when two gravitating particles are

Suppose we've got two still particles of any mass we would like, infinitely far apart.
The total energy here is of course sum of their mass energies.
Now if we bring them together so that one orbits around the other, what would be the total energy of this system? Perhaps, my guess, in such situation where there are only two still particles, without external force, since it is not possible for them to form an orbiting system (please check this for me, but I think it should be true), the change in total energy of the system will be only equal to the work done by the external force.
Is that true?

Suppose we've got two still particles of any mass we would like, infinitely far apart.

Ok.

The total energy here is of course sum of their mass energies.

Not OK. What is mass energy supposed to be?

Now if we bring them together so that one orbits around the other, what would be the total energy of this system?

The potential energy of the masses with respect to each other plus the kinetic energy of the masses in the centre of mass inertial frame plus a possible kinetic energy due to the velocity of the centre of mass plus an arbitrary constant (if we are not using relativity)

Perhaps, my guess, in such situation where there are only two still particles, without external force, since it is not possible for them to form an orbiting system (please check this for me, but I think it should be true), the change in total energy of the system will be only equal to the work done by the external force.
Is that true?

Yes and no. You do not specify how you want to "bring them together". Lets say that they are very far apart but not infinitely far apart. Then they will attract due to gravity and finally collide with a speed that is given by the potential energy at the distance of collision. If one particle has a tiny bit of tangential speed the particles can form a highly elliptical orbit. Not much is necessary, because the particles speed up a lot the closer they get. The energy of this system is conserved. No force is necessary to bring the particles together. If forces are applied then the energy may change.

Just the energy equivalent of their masses.
Is it possible for two still particles finite distance apart to come together and form any orbit at all? Because I don't see how they could get that crucial bit of 'tangential' velocity.

Staff: Mentor

Just the energy equivalent of their masses.
Is it possible for two still particles finite distance apart to come together and form any orbit at all? Because I don't see how they could get that crucial bit of 'tangential' velocity.

They don't acquire that tiny bit of tangential velocity, they have to start with it. If they started out completely at rest relative to one another, so there was absolutely no tangential component to their relative velocity, then they wouldn't form an orbit - they'd just move straight towards each other until they collided. That's easier said than done.

Btw, the key principle at work here is conservation of angular momentum (along with conservation of energy). The sum of the kinetic and potential energy is constant, as is the total angular momentum. The particles will end up in some sort of orbit unless they started with exactly zero angular momentum, which would mean zero tangential velocity.

The energy of the two masses orbiting each other is less than the energy of the masses infinitely apart and not moving. That's because gravity has a negative potential energy. If the masses are bound together by gravity, that means the potential energy plus the kinetic energy is negative. If you take the average over an orbital period for a gravitationally bound pair, the potential energy is twice as large as the kinetic energy and negative thanks to the virial theorem.